Network Policy Targeting: Identification and Known Propensities
Choosing whom to treat in a network requires counting gains to the recipient and to other people. A rule that looks best using direct effects alone can therefore be suboptimal once spillovers are included. The policy-learning problem in (Viviano 2025) first identifies welfare for each candidate rule, then bounds the loss from choosing a rule using estimated welfare.
1. Identify welfare under a rule
Let \(\pi(X_i)\in\{0,1\}\) be an individualized rule. Outcomes depend on own treatment \(D_i\), a specified summary \(T_i\) of neighboring treatments, covariates \(Z_i\), and degree. Write \(T_i(\pi)\) for the exposure induced by applying \(\pi\) to the population.
Let \(I_i(\pi)=1\{D_i=\pi(X_i),T_i=T_i(\pi)\}\), and let \(e_i(\pi)\) be the probability of this exposure event under the experiment, conditional on the required network, covariate, and sampling information. If \(R_i\) indicates experimental participation and \(n_e=\mathbb{E}\sum_iR_i\), the paper identifies conditional welfare by
Identification requires that candidate rules induce exposures supported by the experiment. It also uses the specified participation and assignment mechanisms, consistency, and the restriction describing which assignments affect each outcome.
2. The oracle regret result
Theorem 1. With known exposure probabilities and a bounded outcome adjustment independent of the experiment conditional on baseline information, the empirical welfare maximizer satisfies a bound on conditional expected regret of order
under the paper’s local-dependence, moment, sampling, and policy-class conditions. Here \(\mathcal{N}_n\) is the degree bound used in the theorem, \(\gamma\delta_n\) controls exposure overlap, and \(\Gamma\) bounds the relevant outcome scale. With bounded degree, fixed complexity, and overlap bounded away from zero, this gives the usual \(n_e^{-1/2}\) regret rate. The paper also gives minimax lower bounds for its specified class.
Use the degree convention in the theorem, bounded below by a positive constant greater than one when writing its logarithmic factor. A literal \(\log0\) or \(\log1\) is not a meaningful rate for an empty or degree-one network.
3. The whole regret argument starts with one inequality
Let \(\pi^*\) maximize population welfare and let \(\widehat\pi\) maximize estimated welfare \(\widehat W\). Then
Add and subtract estimated welfare at the population and empirical optimizers. Empirical optimality makes the middle difference nonpositive, leaving two estimation errors. Both are bounded by the same supremum, so the learning problem reduces to controlling welfare estimation uniformly over the policy class.
4. Why exposure overlap can be harder than treatment overlap
Overlap must be checked for the exposure that a rule induces. Moderate individual treatment probabilities can still make a particular treated-neighbor count rare. Those cells receive large inverse weights. Coarsening the summary helps only when outcomes depend on assignments through the coarser summary.
Individualized rules can be deployed using observed characteristics, while the objective still accounts for spillovers. More flexible rules using the whole target adjacency matrix require more network information and a different complexity analysis. The next note handles estimated nuisance functions; the same note also develops trimming and transport.
5. A rule determines a whole exposure pattern
Suppose the exposure summary is the number of treated neighbors. Conditional on the baseline network and covariates,
Assigning treatment to person \(i\) changes that person’s own treatment and the exposure counts used by nearby outcomes. To evaluate a rule, apply it to the relevant neighbors first, then evaluate each outcome contribution under the resulting assignments.
For example, take two connected people and conditional mean \(m_i(d,t)=\tau_i d+\beta_i t\). The total welfare gain from treating only person one is \(\tau_1+\beta_2\), not just \(\tau_1\). Under a one-person budget, comparing only own-treatment effects can choose the wrong recipient. The policy objective sums the effects on every person’s outcome, which is why local decisions are coupled in the optimization.
6. Exact-count overlap can be much weaker than own-treatment overlap
If experimental assignments are independent Bernoulli\((p)\), a unit with \(k\) neighbors has probability
This formula presumes the outcome depends on the neighbor count rather than their identities. At \(p=1/2\), it becomes \(2^{-(k+1)}\binom{k}{t_i}\). A policy inducing all treated neighbors has exponentially small matching probability, while counts near \(k/2\) are much more common.
A full-neighbor-vector exposure requires one exact configuration, whereas a count exposure pools all configurations with that count. This accounts for the binomial coefficient in the latter probability. Pooling improves overlap only when these configurations have the same outcome under the assumed model.
7. The regret decomposition with an approximate optimizer
Let \(\pi^*\) maximize \(W\) over the specified class. Add and subtract empirical welfares:
If the empirical optimization gap is at most \(\epsilon_{\rm opt}\), the middle term is at most that gap. Bounding each remaining term by the same supremum proves
The comparator is the best rule in \(\Pi\). Restricting the class makes uniform estimation easier, but may exclude policies with higher welfare. Enlarging it trades this approximation error against the greater sampling error in comparing more policies.
8. A finite-class concentration argument showing the degree cost
Consider the explanatory special case with every unit participating, a finite class of \(K\) rules, and welfare contributions in an interval of length \(B\). Suppose their dependence graph can be colored with \(\chi\) colors so that contributions within each color class are independent. For a fixed rule, write its centered sum as \(S=\sum_{c=1}^\chi S_c\).
Within each color class, independence gives a product of bounded-variable moment bounds. Hölder’s inequality combines the color classes without assuming they are independent of one another:
Optimizing the exponential tail bound and taking a union bound over rules yields
The resulting uniform error has order \(B\sqrt{\chi\log K/n}\). Large inverse weights increase the range \(B\); overlapping neighborhoods increase the number of colors \(\chi\). The paper replaces this finite-class argument with its VC-class and sampling analysis, producing the stated dependence on degree and overlap.
The concentration bound translates directly into an implementable accuracy statement. With probability at least \(1-\eta\), in the finite-class special case,
This follows by setting \(t=B\sqrt{\chi\log(2K/\eta)/(2n)}\) in the tail bound and multiplying by two in the regret inequality. If outcomes lie in \([0,\Gamma]\) and every relevant exposure has probability at least \(e_0\), an unadjusted inverse-weighted contribution has range at most \(B=\Gamma/e_0\). To make the stochastic part at most \(\varepsilon\), this sufficient bound requires
The square in \(e_0^{-2}\) comes from a range-based concentration argument and can be pessimistic relative to variance-sensitive bounds. Nevertheless, it shows why enlarging the policy class to include extreme neighbor exposures can overwhelm any gain from finer targeting. The comparison is uniform because the maximizing policy was chosen using the estimated welfares themselves.
Uniformity is needed because the selected rule depends on the estimated welfares. A fixed-rule guarantee need not survive selecting whichever estimate is largest. The paper controls this selection through restrictions on the policy class and network sampling model.
9. Why an outcome adjustment can help even with known probabilities
For one exposure cell with probability \(e\) and conditional mean \(m\), take a fixed adjustment \(a\) and score \(a+I(Y-a)/e\). Its expectation is \(m\). If the conditional outcome variance in the cell is \(v\), direct expansion gives
The last expression is minimized at \(a=m\): an accurate outcome adjustment removes variability due to exposure matching, even with known assignment probabilities. The calculation treats that adjustment as fixed under the relevant conditioning. Estimating it from dependent evaluation outcomes requires a separation argument.
The augmented procedure in the learning and transport note permits both nuisance functions to be estimated. Its additional error term records what the oracle analysis leaves out.